I have worked with cellular automata (CA) and would proffer binary CA as a candidate, but I'd be interested to know how other fields answer this question.
It would be really helpful for teaching undergraduates on an interdisciplinary course.
In physics:
- two pendulums on the common suspension. This is the basic model for the chaos theory.
- Van der Pol oscillator (a simple schematics) from the electronics engineering which is a base model for the dissipative chaotic systems.
- 3-bodies problem in celestial mechanics
In mathematics:
- logistic mapping (equation) - the basement of the chaos
- predator/prey equation, maybe. It is quite simple on the first glance
- axiom of choice (well, just kidding)
> logistic mapping (equation) - the basement of the chaos
In discrete time only. Everything is very smooth in continuous time.
> predator/prey equation, maybe.
Rather no than yes... :)
Thank you. I particularly like the double pendulum and the 3-body problem as it appeals to the non-mathematician.
Asked another way, can we find a system that contains/exhibits the features of complexity (feedback, emergence, chaos, etc.) whilst being described by a minimal number of parameters and variables?
I appeal particularly for answers from other fields such as the social sciences, economics and the arts!
> can we find a system that contains/exhibits the features
Yes. Logistics growth equation (aka Ferhulst equation): 1 variable, 1 parameter.
Classical prey-predator model cannot exhibit chaos due to Poincaré-Bendixson theorem (see Nonlinear Dynamics and Chaos 2nd ed. by Strogatz at page 205, for example), since you need a three dimensional phase space.
If you are looking for the simplest continuous complex system with chaotic dynamics, check the Rössler system (same book, page 383), it has just one nonlinear term in three equations.
well, it leads us into a discussion about the definitions of "mathematical" and "non-mathemetical" systems.
I support the view that mathematics is rather experimental science by its nature (and origins), so it can be seen as a language which describes the "systems". The "systems" is in quotes because this therm should be better defined in the context of the discussion.
With the current (intuitive) meaning of the "system", we can say that any "system" is a kind of "mathematical" one.
O! I know a system of chaos with non-mathematical background. But I cannot tell it: it sould be considered as something politically incorect... :)
Why? Rich inner world has never been incorrect by any means (but mathematics).
No, no! I do not want to cause any international calamities... :):)
I agree that it is difficult to identify chaos in a 'non-mathematical' system. Likewise, I agree that any system has the potential to be described mathematically!
I would then return to my original question (deliberately vague and succinct), with some additional terms:
In your field, what would you say is the simplest complex system?
(By this I mean that mathematicians, engineers, biologists, economists, ... each may have their own notion of a 'complex system', and so it would be interesting to compare these ideas.)
> n your field, what would you say is the simplest complex system?
In my field that is the distribution of genetic entities in the space of oligonucleotide frequencies. And (rarely) the dynmaics of some biological communities.
The simplicity is, of course, relative...
Cellular automata are containing the simplest complex systems capable of universal computation as proven by their equivalence with Turing machine. One superb example is the Game of Life. It is possible to build computers using this rule. There are other similar rules in existence.
Now, toward your question. Biological systems are made from many massively parallel simple subsystems that are networked. I recommend looking in this direction for other mathematical examples like self-assembly, protein binding, and similar.
There is an existing Nagel-Shrekenberg model that is implemented in CAs and perfectly describes traffic jam creation. There are existing YouTube videos that will illustrate the issue very well, same for sandpile models.
Self-organized criticality models of sandpile behavior, earthquakes, volcano activities, stock market behavior, stellar behavior, and many other examples good to mention. Those are again best to implement in CA-like models.
Then, it would be good to dive into coupled mass-spring models. They are easy to describe, yet they express surprisingly high complexity.
All of the above-mentioned models fail into the category of massive parallel models. They can be described by many approaches but CAs provide usually the simplest description from all known.
> Jiří Kroc
What kind of complexity could be found in Game of Life? It is a finite object, and the notiuon of complexity for finite objects in complex itself... :)
@Michael Sadovski
The 'Game of Life' is fascinating because it is proven to be Turing machine equivalent. In other words, it can compute what is computable. There are people who had designed AND, OR, and NOT gates using glider guns and gliders. This enabled them to construct computers within the GoL.
It is true, those simulations are slow but on some cheap abundant hardware or wetware it is OK.
The GoL opened doors into the world of self-organizing systems, emergence, self-replicating structures, and such in mathematical sense.
Emergence is the native language of biological structures. It is the language our bodies use to function. That is why The GoL and research triggered by it are important.
When you are interested about those issues, you can check out two projects that are dealing with those issues:
https://www.researchgate.net/project/Complexity-in-Medicine-Practical-Problems-Their-Definitions-Models-and-Solutions
and
https://www.researchgate.net/project/Complexity-Digest--Toolkit-Containing-Root-Research-Results-Articles-Books-and-Software--Covers-Emergence-Self-Organzation-and-Beyond
2 Jiri Kroc: I meant Zvonkin-Levin approach: А. К. Звонкин, Л. А. Левин, Сложность конечных объектов и обоснование понятий информации и случайности с помощью теории алгоритмов, УМН, 1970, том 25, выпуск 6(156), 85–127
Soviet Math Uspekhi, to the best of my knowledge is the title of the journal in English.
It depends on the definition of what is the "simplest" in this case. For example, it could be "simplest" by behavior (like computation in CA), or it could be "simplest" by the number of elements in the complex system.
In the latter sense, that would be a complex system, which (a) consist of two elements, (b) new elements emerge over time, and (c.) based on characteristics of existing elements nobody could predict (compute) when and which next element will emerge in the system.
Going back to the original question, watch the following video (animations start at 1:30 min):
"The John H. Conway's 'Game of Life' (GoL) Cellular Automaton":
https://youtu.be/gc42D2sdL8I
and feel free to discuss what you see there. Animations gives the far best insights without going into details of programming and mathematics.
Those simulations can speak to all of us: mathematicians, physicists, biologists, computer scientists, and all enthusiasists. Every one can see there something else. Enjoy.
PS. Code in Python is open-source:-)
Thank you for this. I agree, simulations like these appeal to people from a wide variety of abilities and backgrounds. At the London Interdisciplinary School (https://www.lis.ac.uk/), simulations and Python coding forms a big part our quantitative teaching for such an interdisciplinary degree.
Here, using this animation, it is very easy explain arising of the second-order emergents from almost any random condition in the cellular automaton, which has the fixed, shown neighborhood and purely deterministic updating rule.
This observation has far reaching consequences that are going to be discussed, beside many other aspects of such simulations, in the future paper. Stay tuned to see other similarly exciting results.
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